About once a year I have to take apart the bathroom sink trap and flush a "clog" of slimy, black, gelatinous goop out of it. This only happens in my bathroom sink, and there is no hair in it, which leads me to think the gelatinous mess has something to do with listerine or mentadent, or some combination thereon. The blockage does not build up in the trap, but in the straight-pipe above it, just underneath the sink. The drainage-slowdown is indeed a long-building process, hence the need to clear the blockage only about once a year, which is (of course) a dirty, smelly affair.

1) If you're a listerine user, do you notice anything like this?

2) Is there any other way to remove this clog? Can I pour boiling water down the pipe in an attempt to melt it away?

3) If I leave it long enough, will it turn into some kind of Japanese monster that Godzilla will have to come kill?

It's everything. Hair, soap scum, bacteria, Think of your sink more completely in the context of the yearly load it's gotta handle.
It's a lot. Anything heavy that doesn't get washed down ends up in the traps

Dilute Bleach yah, hot water yah, Vinegar yah,
Drano - - not if you have any metal pipes don't you dare.

If it's all PVC then fine, but what about that last stage that is underground and the MOST expensive to repair - - on the way to the town sewer or your septic tank? Chances are you got iron in there somewhere unless it's brand new construction. And even then - - unless you know. Be careful with anything that is intended to sit in place.

It's bacteria. using the Listerine has probably created a strain of bacteria that is immune to the Listerine, Bleach is the only 100% sure killer of bacteria, ( that you would have access to).
Don't dilute, use 1-2 cups, let it sit for at least 30 min, and flush with boiling water.

Look at this:
what product(s) do you normally use at that sink that you don't use at any other sink in the house? Some special cosmetic or chemical formulation ? that goes down there and maybe interacts with some other, maybe organic thing that also passes through that place? that maybe unintentionally winds up as a slime -generating reaction? between biochemical reaction to industrial chemistry?

I do a white vinegar, baking soda cleanse, once a week. It foams up the scum and helps to break it up. I, in addition, pour a small bottle of Clorox, down all of the drains and toilet, one a month. Pour it in and leave it for at least 1/2 hour, then flush with water for a while.

we had the same problem at my last place, also just in one sink (no listerine or anything, just Tom's Of Maine toothpaste, soap, and nothing else).

Yuck. . If you have more than one bathroom sink in the house and only one does it, it's probably just that one is 'infected' with the offending biofilm- you can't kill all of the nasty stuff completely, so it re-grows, and others don't have the same issue. Yet.

you might want to research live bacteria cleaners. ‘Plumb-It’ and ‘Drain-Care’ are just some of the products that use all-natural, environmentally safe biological formulas that blend non-toxic live bacteria to consume any build up of grease, soaps, fats oils and foods and while it won’t remove the problem right away – takes time to act – but if you use it regularly, the bacteria coats the inside of the drain line system, using a natural way to keep drain lines, grease traps and septic tanks working naturally by turning waste into harmless carbon dioxide and water. reduces any drain smells too. most nonprofit consumer groups that promote environmental sustainability okay these types of products over harsh chemicals such as bleach etc. i’ve used them for years and they have made my plumbing trouble free.